Intuitive Development
• Intuitively, the probability of an event a could be defined as:
Where N(a) is the number that event a happens in n trialsWhere N(a) is the number that event a happens in n trials
More Formal:
• is the Sample Space:– Contains all possible outcomes of an experiment
• 2 is a single outcome• A 2 is a set of outcomes of interest
Independence
• The probability of independent events A, B and C is given by:
P(ABC) = P(A)P(B)P(C)
A and B are independent, if knowing that A has happened A and B are independent, if knowing that A has happened does not say anything about B happeningdoes not say anything about B happening
Random Variables
• A (scalar) random variable X is a function that maps the outcome of a random event into real scalar values
X(X())
Random Variables Distributions
• Cumulative Probability Distribution (CDF):
• Probability Density Function (PDF):Probability Density Function (PDF):
Uniform Distribution
• A R.V. X that is uniformly distributed between x1 and x2 has density function:
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Statistical Characterizations
• Expectation (Mean Value, First Moment):
•Second Moment:Second Moment:
Mean Estimation from Samples
• Given a set of N samples from a distribution, we can estimate the mean of the distribution by:
Variance Estimation from Samples
• Given a set of N samples from a distribution, we can estimate the variance of the distribution by:
Measuring Noise
• Noise Amount: SNR = s/ n
• Noise Estimation: – Given a sequence of images I0,I1, … IN-1
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Good estimators
Data values z are random variablesA parameter describes the distributionWe have an estimator z) of the unknown parameter
If E(z) or E(z) ) = E(the estimatorz) is unbiased
Least Squares (LS)
If errors only in b
Then LS is unbiased
But if errors also in A (explanatory variables)
Least Squares (LS)
biasLarger variance in A,,ill-conditioned A,u oriented close to the eigenvector of the smallest eigenvalue increase the biasGenerally underestimation
(a) (b)
Estimation of optical flow
(a) Local information determines the component of flow perpendicular to edges(b) The optical flow as best intersection of the flow constraints is biased.
Optical flow
• One patch gives a system:
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